Two-log-convexity of the Catalan-Larcombe-French sequence

Abstract: The Catalan-Larcombe-French sequence ${P_n}{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence ${P^2_n-P{n-1}P_{n+1}}_{n\geq 1}$ are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.